Complex and Nonlocal Systems

该提案涉及从复杂流体到社交网络等各种环境中出现的复杂和非本地系统。研究微观对象影响其宏观环境的混合系统是非线性科学的基本组成部分。该项目考虑了非线性偏微分方程组模型，其中微观部分由源自基本随机方程的动力学方程和宏观流体方程来建模。该方案针对弱解、混合、长时间行为和统计解的奇性形成、存在唯一性或缺乏奇性等基本问题。该提案还涉及对主动网络的研究，主动网络是响应网络依赖过程而演变的自然或人工网络。该方案的总体目标是研究非局部相互作用和反馈对演化复杂系统的动力学效应。该方案解决了复杂非局部系统的建模、计算和理论中出现的问题。这样的系统发生在各种各样的环境中，从复杂的物质，如血液或细胞内液体，到复杂的关系系统，如社会网络。复杂物质的特征是存在由非常小的物体组成的复杂的、相互作用的网络，这些网络被嵌入、影响并反过来影响基质或溶剂。对这些相互作用的基本了解对于在应用方面取得进展至关重要，例如制造具有极端性能的材料或在细胞水平上输送药物。带反馈的信息通过这样的网络传播对于基本生物过程的研究和人工网络的研究都是非常重要的。该提案的目的是揭示从本地和非本地相互作用中出现的系统规模质变的基本机制。

The proposal addresses complex and nonlocal systems, arising in a variety of contexts, ranging from complex fluids to social networks. The study of mixed systems in which microscopic objects influence their macroscopic environment is a fundamental component of nonlinear science. The project considers nonlinear PDE models in which the microscopic part is modelled by kinetic equations derived from underlying stochastic equations, coupled with macroscopic fluid equations. The proposal aims at basic questions of singularity formation, existence and uniqueness or lack thereof for weak solutions, mixing, long time behavior and statistical solutions. The proposal is concerned also with the study of active networks, natural or artificial networks that evolve in response to network-dependent processes. The overarching goal of the proposal is to study the dynamical effects of nonlocal interaction and feedback on evolving complex systems.The proposal addresses problems that arise in the modeling, computation and theory of complex nonlocal systems. Such systems occur in a great variety of circumstances, ranging from complex matter, such as blood or intracellular fluids, to complex relational systems, such as social networks. Complex matter is characterized by the presence of complex, interacting networks of very small objects being embedded, influenced, and in turn influencing a matrix or a solvent. The basic understanding of these interactions is crucial for progress in applications such as the manufacturing of materials with extreme properties or the delivery of drugs at the cellular level. Diffusion of information with feedback through such networks is of great importance for the study of basic biological processes and also for the study of artificial networks. The aim of the proposal is to uncover fundamental mechanisms of system-size qualitative changes emerging from local and nonlocal interactions.